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131
EIGENVALUES AND EXPANDERS
 COMBINATORICA
, 1986
"... Linear expanders have numerous applications to theoretical computer science. Here we show that a regular bipartite graph is an expander ifandonly if the second largest eigenvalue of its adjacency matrix is well separated from the first. This result, which has an analytic analogue for Riemannian mani ..."
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Cited by 404 (19 self)
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Linear expanders have numerous applications to theoretical computer science. Here we show that a regular bipartite graph is an expander ifandonly if the second largest eigenvalue of its adjacency matrix is well separated from the first. This result, which has an analytic analogue for Riemannian manifolds enables one to generate expanders randomly and check efficiently their expanding properties. It also supplies an efficient algorithm for approximating the expanding properties of a graph. The exact determination of these properties is known to be coNPcomplete.
Concentration Of Measure And Isoperimetric Inequalities In Product Spaces
, 1995
"... . The concentration of measure phenomenon in product spaces roughly states that, if a set A in a product# N of probability spaces has measure at least one half, "most" of the points of# N are "close" to A. We proceed to a systematic exploration of this phenomenon. The meaning ..."
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Cited by 383 (4 self)
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. The concentration of measure phenomenon in product spaces roughly states that, if a set A in a product# N of probability spaces has measure at least one half, "most" of the points of# N are "close" to A. We proceed to a systematic exploration of this phenomenon. The meaning of the word "most" is made rigorous by isoperimetrictype inequalities that bound the measure of the exceptional sets. The meaning of the work "close" is defined in three main ways, each of them giving rise to related, but di#erent inequalities. The inequalities are all proved through a common scheme of proof. Remarkably, this simple approach not only yields qualitatively optimal results, but, in many cases, captures near optimal numerical constants. A large number of applications are given, in particular to Percolation, Geometric Probability, Probability in Banach Spaces, to demonstrate in concrete situations the extremely wide range of application of the abstract tools. AMS Classification numbers: Primary 60E15, 28A35, 60G99; Secondary 60G15, 68C15. Typeset by A M ST E X 1 2 M. TALAGRAND Table of Contents I.
The Laplacian spectrum of graphs
 Graph Theory, Combinatorics, and Applications
, 1991
"... Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, m ..."
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Cited by 226 (2 self)
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Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidthtype parameters of a graph. Some new results and generalizations are added. † This article appeared in “Graph Theory, Combinatorics, and Applications”, Vol. 2,
On Talagrand's Deviation Inequalities For Product Measures
, 1996
"... We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M. ..."
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Cited by 113 (0 self)
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We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M.
A Riemannian interpolation inequality à la Borell, Brascamp and Lieb
, 2001
"... A concavity estimate is derived for interpolations between L¹(M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell, Brascamp and Lieb that it generalizes from Euclidean space. Due to the curvature of the manifold, the new Riemannian ..."
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Cited by 87 (9 self)
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A concavity estimate is derived for interpolations between L¹(M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell, Brascamp and Lieb that it generalizes from Euclidean space. Due to the curvature of the manifold, the new Riemannian versions of these theorems incorporate a volume distortion factor which can, however, be controlled via lower bounds on Ricci curvature. The method uses optimal mappings from mass transportation theory. Along the way, several new properties are established for optimal mass transport and interpolating maps on a Riemannian manifold.
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
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Eigenvalues in combinatorial optimization
, 1993
"... In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey. ..."
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Cited by 44 (0 self)
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In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey.
On the role of convexity in isoperimetry, spectralgap and concentration
 Invent. Math
"... We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the apriori weakest requirement that Lipschitz functions have arbitrarily slow uniform taildecay, are all quantitativ ..."
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Cited by 44 (12 self)
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We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the apriori weakest requirement that Lipschitz functions have arbitrarily slow uniform taildecay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz’ya, Cheeger, Gromov– Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are “onaverage ” Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the “worst ” subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne–Weinberger, Li–Yau, Kannan– Lovász–Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semigroup following Bakry–Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ∞) curvaturedimension condition of BakryÉmery. 1
ON HOEFFDING’S INEQUALITIES
, 2004
"... ... 13–30], several inequalities for tail probabilities of sums Mn = X1 +···+ Xn of bounded independent random variables Xj were proved. These inequalities had a considerable impact on the development of probability and statistics, and remained unimproved until 1995 when Talagrand [Inst. Hautes Étud ..."
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Cited by 40 (4 self)
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... 13–30], several inequalities for tail probabilities of sums Mn = X1 +···+ Xn of bounded independent random variables Xj were proved. These inequalities had a considerable impact on the development of probability and statistics, and remained unimproved until 1995 when Talagrand [Inst. Hautes Études Sci. Publ. Math. 81 (1995a) 73–205] inserted certain missing factors in the bounds of two theorems. By similar factors, a third theorem was refined by Pinelis [Progress in Probability 43 (1998) 257–314] and refined (and extended) by me. In this article, I introduce a new type of inequality. Namely, I show that P{Mn ≥ x}≤cP{Sn ≥ x}, wherecis an absolute constant and Sn = ε1 +···+εn is a sum of independent identically distributed Bernoulli random variables (a random variable is called Bernoulli if it assumes at most two values). The inequality holds for those x ∈ R where the survival function x ↦ → P{Sn ≥ x} has a jump down. For the remaining x the inequality still holds provided that the function between the adjacent jump points is interpolated linearly or loglinearly. If it is necessary, to estimate P{Sn ≥ x} special bounds can be used for binomial probabilities. The results extend to martingales with bounded differences. It is apparent that Theorem 1.1 of this article is the most important. The inequalities have applications to measure concentration, leading to results of the type where, up to an absolute constant, the measure concentration is dominated by the concentration in a simplest appropriate model, such results will be considered elsewhere.